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In mathematics, the infinite series 1 / 2 − 1 / 4 + 1 / 8 − 1 / 16 + ⋯ is a simple example of an alternating series that converges absolutely. It is a geometric series whose first term is 1 / 2 and whose common ratio is − 1 / 2, so its sum is. Mar 20, 2013 Download smartsuite for free. UCSC SMART suite controls and monitors storage devices using the Self-Monitoring, Analysis and Reporting Technology System(S.M.A.R.T.) build into ATA and SCSI Hard Drives. This is used to check the reliability of the hard drive and predict drive failures. ‎Pages is the most beautiful word processor you’ve ever seen on a mobile device. Start with an Apple-designed template to instantly create gorgeous reports, digital books, resumes, posters and more. Or use a blank document and create your own design. Easily add images, movies, audio, charts and shap.

First six summands drawn as portions of a square.

The geometric series on the real line.

In mathematics, the infinite series1/2 + 1/4 + 1/8 + 1/16 + ··· is an elementary example of a geometric series that converges absolutely.

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There are many different expressions that can be shown to be equivalent to the problem, such as the form: 2⁻¹ + 2⁻² + 2⁻³ + ..

The sum of this series can be denoted in summation notation as:

{displaystyle {frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots =sum _{n=1}^{infty }left({frac {1}{2}}right)^{n}={frac {frac {1}{2}}{1-{frac {1}{2}}}}=1.}

Proof[edit]

As with any infinite series, the infinite sum

{displaystyle {frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots }

is defined to mean the limit of the sum of the first n terms

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{displaystyle s_{n}={frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+{frac {1}{16}}+cdots +{frac {1}{2^{n-1}}}+{frac {1}{2^{n}}}}

as n approaches infinity.

Multiplying s_n by 2 reveals a useful relationship:

{displaystyle 2s_{n}={frac {2}{2}}+{frac {2}{4}}+{frac {2}{8}}+{frac {2}{16}}+cdots +{frac {2}{2^{n}}}=1+left[{frac {1}{2}}+{frac {1}{4}}+{frac {1}{8}}+cdots +{frac {1}{2^{n-1}}}right]=1+left[s_{n}-{frac {1}{2^{n}}}right].}

Subtracting s_n from both sides,

{displaystyle s_{n}=1-{frac {1}{2^{n}}}.}

As n approaches infinity, s_ntends to 1.

History[edit]

Zeno's paradox[edit]

This series was used as a representation of many of Zeno's paradoxes, one of which, Achilles and the Tortoise, is shown here.^[1] In the paradox, the warrior Achilles was to race against a tortoise. The track is 100 meters long. Achilles could run at 10 m/s, while the tortoise only 5. The tortoise, with a 10-meter advantage, Zeno argued, would win. Achilles would have to move 10 meters to catch up to the tortoise, but by then, the tortoise would already have moved another five meters. Achilles would then have to move 5 meters, where the tortoise would move 2.5 meters, and so on Zeno argued that the tortoise would always remain ahead of Achilles.

The Eye of Horus[edit]

The parts of the Eye of Horus were once thought to represent the first six summands of the series.^[2]

References[edit]

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^Wachsmuth, Bet G. 'Description of Zeno's paradoxes'. Archived from the original on 2014-12-31. Retrieved 2014-12-29.
^Stewart, Ian (2009). Professor Stewart's Hoard of Mathematical Treasures. Profile Books. pp. 76–80. ISBN978 1 84668 292 6.

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